Wick product

In probability theory, the Wick product is a particular way of defining an adjusted product of a set of random variables. In the lowest order product the adjustment corresponds to subtracting off the mean value, to leave a result whose mean is zero. For the higher order products the adjustment involves subtracting off lower order (ordinary) products of the random variables, in a symmetric way, again leaving a result whose mean is zero. The Wick product is a polynomial function of the random variables, their expected values, and expected values of their products.

The definition of the Wick product immediately leads to the Wick power of a single random variable and this allows analogues of other functions of random variables to be defined on the basis of replacing the ordinary powers in a power-series expansions by the Wick powers. The Wick powers of commonly-seen random variables can be expressed in terms of special functions such as Bernoulli polynomials or Hermite polynomials.

The Wick product is named after physicist Gian-Carlo Wick, cf. Wick's theorem.

Definition

Assume that X1, ..., Xk are random variables with finite moments. The Wick product

⟨ X 1 , … , X k ⟩ {\displaystyle \langle X_{1},\dots ,X_{k}\rangle \,}

is a sort of product defined recursively as follows:[citation needed]

⟨ ⟩ = 1 {\displaystyle \langle \rangle =1\,}

(i.e. the empty product—the product of no random variables at all—is 1). For k ≥ 1, we impose the requirement

∂ ⟨ X 1 , … , X k ⟩ ∂ X i = ⟨ X 1 , … , X i − 1 , X ^ i , X i + 1 , … , X k ⟩ , {\displaystyle {\partial \langle X_{1},\dots ,X_{k}\rangle \over \partial X_{i}}=\langle X_{1},\dots ,X_{i-1},{\widehat {X}}_{i},X_{i+1},\dots ,X_{k}\rangle ,}

where X ^ i {\displaystyle {\widehat {X}}_{i}} means that Xi is absent, together with the constraint that the average is zero,

E ⁡ ⟨ X 1 , … , X k ⟩ = 0. {\displaystyle \operatorname {E} \langle X_{1},\dots ,X_{k}\rangle =0.\,}

Equivalently, the Wick product can be defined by writing the monomial X 1 … X k {\displaystyle X_{1}\dots X_{k}} as a "Wick polynomial":

X 1 … X k = ∑ S ⊆ { 1 , … , k } E ⁡ ( ∏ i ∉ S X i ) ⋅ ⟨ X i : i ∈ S ⟩ {\displaystyle X_{1}\dots X_{k}=\sum _{S\subseteq \left\{1,\dots ,k\right\}}\operatorname {E} \left(\textstyle \prod _{i\notin S}X_{i}\right)\cdot \langle X_{i}:i\in S\rangle \,} ,

where ⟨ X i : i ∈ S ⟩ {\displaystyle \langle X_{i}:i\in S\rangle } denotes the Wick product ⟨ X i 1 , … , X i m ⟩ {\displaystyle \langle X_{i_{1}},\dots ,X_{i_{m}}\rangle } if S = { i 1 , … , i m } {\displaystyle S=\left\{i_{1},\dots ,i_{m}\right\}} . This is easily seen to satisfy the inductive definition.

Examples

It follows that

⟨ X ⟩ = X − E ⁡ X , {\displaystyle \langle X\rangle =X-\operatorname {E} X,\,}
⟨ X , Y ⟩ = X Y − E ⁡ Y ⋅ X − E ⁡ X ⋅ Y + 2 ( E ⁡ X ) ( E ⁡ Y ) − E ⁡ ( X Y ) , {\displaystyle \langle X,Y\rangle =XY-\operatorname {E} Y\cdot X-\operatorname {E} X\cdot Y+2(\operatorname {E} X)(\operatorname {E} Y)-\operatorname {E} (XY),\,}
⟨ X , Y , Z ⟩ = X Y Z − E ⁡ Y ⋅ X Z − E ⁡ Z ⋅ X Y − E ⁡ X ⋅ Y Z + 2 ( E ⁡ Y ) ( E ⁡ Z ) ⋅ X + 2 ( E ⁡ X ) ( E ⁡ Z ) ⋅ Y + 2 ( E ⁡ X ) ( E ⁡ Y ) ⋅ Z − E ⁡ ( X Z ) ⋅ Y − E ⁡ ( X Y ) ⋅ Z − E ⁡ ( Y Z ) ⋅ X − E ⁡ ( X Y Z ) + 2 E ⁡ ( X Y ) E ⁡ Z + 2 E ⁡ ( X Z ) E ⁡ Y + 2 E ⁡ ( Y Z ) E ⁡ X − 6 ( E ⁡ X ) ( E ⁡ Y ) ( E ⁡ Z ) . {\displaystyle {\begin{aligned}\langle X,Y,Z\rangle =&XYZ\\&-\operatorname {E} Y\cdot XZ\\&-\operatorname {E} Z\cdot XY\\&-\operatorname {E} X\cdot YZ\\&+2(\operatorname {E} Y)(\operatorname {E} Z)\cdot X\\&+2(\operatorname {E} X)(\operatorname {E} Z)\cdot Y\\&+2(\operatorname {E} X)(\operatorname {E} Y)\cdot Z\\&-\operatorname {E} (XZ)\cdot Y\\&-\operatorname {E} (XY)\cdot Z\\&-\operatorname {E} (YZ)\cdot X\\&-\operatorname {E} (XYZ)\\&+2\operatorname {E} (XY)\operatorname {E} Z+2\operatorname {E} (XZ)\operatorname {E} Y+2\operatorname {E} (YZ)\operatorname {E} X\\&-6(\operatorname {E} X)(\operatorname {E} Y)(\operatorname {E} Z).\end{aligned}}}

Another notational convention

In the notation conventional among physicists, the Wick product is often denoted thus:

: X 1 , … , X k : {\displaystyle :X_{1},\dots ,X_{k}:\,}

and the angle-bracket notation

⟨ X ⟩ {\displaystyle \langle X\rangle \,}

is used to denote the expected value of the random variable X.

Wick powers

The nth Wick power of a random variable X is the Wick product

X ′ n = ⟨ X , … , X ⟩ {\displaystyle X'^{n}=\langle X,\dots ,X\rangle \,}

with n factors.

The sequence of polynomials Pn such that

P n ( X ) = ⟨ X , … , X ⟩ = X ′ n {\displaystyle P_{n}(X)=\langle X,\dots ,X\rangle =X'^{n}\,}

form an Appell sequence, i.e. they satisfy the identity

P n ′ ( x ) = n P n − 1 ( x ) , {\displaystyle P_{n}'(x)=nP_{n-1}(x),\,}

for n = 0, 1, 2, ... and P0(x) is a nonzero constant.

For example, it can be shown that if X is uniformly distributed on the interval [0, 1], then

X ′ n = B n ( X ) {\displaystyle X'^{n}=B_{n}(X)\,}

where Bn is the nth-degree Bernoulli polynomial. Similarly, if X is normally distributed with variance 1, then

X ′ n = H n ( X ) {\displaystyle X'^{n}=H_{n}(X)\,}

where Hn is the nth Hermite polynomial.

Binomial theorem

( a X + b Y ) ′ n = ∑ i = 0 n ( n i ) a i b n − i X ′ i Y ′ n − i {\displaystyle (aX+bY)^{'n}=\sum _{i=0}^{n}{n \choose i}a^{i}b^{n-i}X^{'i}Y^{'{n-i}}}

Wick exponential

⟨ exp ⁡ ( a X ) ⟩   = d e f   ∑ i = 0 ∞ a i i ! X ′ i {\displaystyle \langle \operatorname {exp} (aX)\rangle \ {\stackrel {\mathrm {def} }{=}}\ \sum _{i=0}^{\infty }{\frac {a^{i}}{i!}}X^{'i}}

References

  • Wick Product Springer Encyclopedia of Mathematics
  • Florin Avram and Murad Taqqu, (1987) "Noncentral Limit Theorems and Appell Polynomials", Annals of Probability, volume 15, number 2, pages 767—775, 1987.
  • Hida, T. and Ikeda, N. (1967) "Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral". Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66). Vol. II: Contributions to Probability Theory, Part 1 pp. 117–143 Univ. California Press
  • Wick, G. C. (1950) "The evaluation of the collision matrix". Physical Rev. 80 (2), 268–272.